Quantitative Techniques Comprehensive Mock Examinations

Part III: Comprehensive Mock Examinations

QUESTIONS - MOCKS

PAPER 1

Time allowed: 3 hours

Answer any THREE questions in SECTION I and TWO questions in SECTION II. Marks

allocated to each question are shown at the end of the question. Show all your workings.

SECTION I

QUESTION ONE

a) Write down short notes on: i) Final demand. (2marks) ii) Technical coefficients. (2marks) iii) Closed model versus open model. (3marks)

b) Three industries, packaging P, Bakery B, and Flour F are related with the intermediate demand matrix below.

Output industry

Input Industry

P B F

P 0 5 0 1 0 1

B 0 2 0 6 0 2

F 0 1 0 2 0 6

The final demand for P, B and F are:

D =

Required:

20

20

50

i) Determine the total output for products of industries P, B and F. (10 marks) ii) Comment on the total output for industries. (3 marks)

QUESTION TWO a) Define the following terms as used in measures of dispersion:

i) Platykurtic versus leptokurtic and mesokurtic. (6marks) ii) Coefficient of skewness. (2marks) iii) Quartile deviation. (2marks)

QUANTITATIVE TECHNIQUES

112 Questions - Mocks

b) A survey by Marketing Society here in Kenya found out the following buying habits of household air conditioners considered a luxury.

Gross income per month

Up to 5,000

5,001 - 10,000

10,001 - 15,000

15,001 - 20,000

20,001 - 25,000

25,001 - 30,000

30,001 - 35,000

35,001 and above.

Number of consumers who buy Air conditioner

2

15

25

28

33

35

38

40

Required: i) Calculate the arithmetic mean, median and mode. (5 marks) ii) Does the survey show that the luxury item is consumed by high-income earners? Show your answer

by calculating appropriate measure of skewness. (5 marks)

QUESTION THREE a) In connection with probability, define the following terms

i) Compound event (2 marks) ii) Mutually exclusive events (2 marks) iii) Collective exhaustive events (2 marks) iv) Equally likely events (2 marks) v) Conditional probability (2 marks)

b) Given the following distribution of wages for 500 technicians in motor companies in industrial area.

Wages in KSh 3000-4000 4000-5000 5000-6000 6000-7000 7000-8000 8000-9000

No. of technicians 30 40 250 100 60 20

A technician is chosen from the above group. What is the probability that his wages are? i) Under KSh 5,000 (4 marks) ii) Above KSh 6,000 (4 marks) iii) Between KSh 5,000 and KSh 6,000 (2 marks)

QUESTION FOUR a) Define the goodness of fit test. How is it applied in accounting? (5 marks) b) A research studying the role of stress and its implication on personal life in respect of job change over by

low cadre staff, came up with the following data. It relates to 30 firms over 3-year period

No. of people changing jobs in 0 1 2 3 4 5 6 7 8 9 a year

Observed frequency 8 18 19 20 16 12 8 4 3 2

By fitting a Poisson distribution to get expected frequency, test its goodness of fit. (15 marks)

QUESTION FIVE a) With reference to linear regression define the following terms:

i) Scatter diagram. ii) Bivariate distribution. iii) Positive correlation. iv) Confidence interval.

v) Auto correlation. (10 marks)

SHAHIHI

Comprehensive Mock Examinations 113

b) The following data relates business turnover and staff of fast moving consumer goods company EAI Ltd:

Year 1993 1994 1995 1996 1997 1998 1999 2000

Business turnover in 45 50 60 75 80 110 150 170 Millions of shillings

Staff 2,600 3,000 3,100 3,530 3,850 4,300 5,870 7,150

Required: i) Fit an appropriate regression equation. (8 marks) ii) Estimate the staff requirement when business turnover reaches Sh.200 Million. (2 marks)

SECTION II

QUESTION SIX a) What are the underlying assumptions in linear programming? (5 marks) b) Zadock ltd. manufactures various kinds of furniture namely ordinary chairs, baby cots, and executive

chairs. All the products use raw material (wood), machine time (lathe) and manual labour (technicians). The requirements for the furniture are summarised as below.

wood in Cubic Lathe time in Technician time meters hours in hours

Ordinary chair 10 3 2 Baby cots 12 5 3 Executive chairs 20 6 5

The estimated profits for each kind of furniture are Sh. 26, Sh. 35 and Sh. 50 respectively. There is a limitation of capacity, in that only 300 cubic meters of wood, 118 hours of lathe time and 90 hours of technician time are available.

This problem was solved using a spreadsheet and the following was obtained.

Target Cell (Max)

Name Final Value

Z 856

X1 6

X2 20

X3 0

Constraints

Name Cell Value Status Slack

wood 300 Binding 0

lathe 118 Binding 0

technician 72 Not Binding 18

X1 6 Not Binding 6

X2 20 Not Binding 20

X3 0 Binding 0

Variables

Final Reduced

Name Value Gradient

X1 6 0

X2 20 0

X3 0 -2



Constraints

Final Dual

Name Value Price

wood 300 1.78571428

lathe 118 2.71428571

technician 72 0

Coefficients

Adjustable Lower Target Upper Target

Name Value Limit Result Limit Result

X1 6 -3.5972E-14 700 6 856

X2 20 0 156 20 856

X3 0 0 856 0 856

Required: i) Formulate the linear programming problem into standard form for input to the computer

spreadsheet (5 marks) ii) What are the units of each kind of furniture to be produced to maximise profits. What is the

maximum profit? (5 marks) iii) Is there any resource that is not used up? For the used up resource if any, how much would you pay

for one additional unit. (5 marks)

QUESTION SEVEN a) Define the following as used in game theory:

i) N - person game. (1mark) ii) Saddle point. (2marks) iii) Strategy. (2marks) iv) Rules of dominance. (3marks)

b) Peter and Ngomongo are second hand dealers of electronic equipment and shoes along Baricho road. Due to hard economic times and restriction by City Council, sales have been decreasing. Each of them came up with strategies to expand into the other business (they have monopoly now). Each of them knows what the other is considering thus influences each others decision. Peter had the following matrix of profit per day.

Ngomongo

Expand Don‟t expand

Peter Expand 1000 0

Don‟t expand -500 500

Required: i) Interpret the matrix. (4marks) ii) Solve the game to determine the average winning (loss) each cousin would have. (8marks)

SHAHIHI


QUESTION EIGHT Vick Press Ltd is planning to expanding into offering security systems in people‟s homes scared of security. The following are activities and associated costs for the expansion:

Normal Crash

Activity Predecessor Time Cost Time Cost weeks Sh. weeks Sh.

A - 10 10,000 7 12,000

B A 35 50,000 33 52,000

C A 4 7,000 3 8,000

D C 25 20,000 25 26,000

E B,D 5 5,000 4 4,500

F C 2 4,000 2 4,000

G F 4 30,000 4 30,000

H G 2 15,000 1 25,000

I E,H,L 1 4,000 1 4,000

J - 8 12,000 4 24,000

K - 12 24,000 10 20,000

L J,K 4 6,000 2 2,000

M J 8 10,000 6 8,000

Required: i) Determine the critical path. (8marks) ii) Determine the minimum time and minimum cost for the networks. (7marks) iii) Given that, for every delay beyond 40th week, there is share of Sh.1,000 per week loss of profit. Is it

advisable to crash the project from 51 to 45 weeks? Why? (5marks)



PAPER 2

Time Allowed: 3 hours

Answer any THREE questions in SECTION ONE and TWO questions in SECTION II. Marks allocated to each question are shown at the end of the question. Show all your workings.

SECTION I

QUESTION ONE a) Kiko Manufacturing Ltd. wants to take a decision of introducing a new soap Kikope. The cost for introducing Kikope (Initial advertising, promotion and fixed cost for one year of production) is estimated at Ksh.30,000. The variable cost per bar of soap is Ksh.30 and the expected selling price is Ksh.50.

Required: Draw the cost, revenue and profit functions and from it:

i) Determine the break-even level of production. (3 marks) ii) Determine the profit on the sale of 2,500 soap bars. (2 marks)

b) Caterpillar Company Ltd. produces generators and drilling engines. The revenue function that describes total revenue for sales in a particular quarter of these engines is given by:

R = 8x + 5y + 2xy - x2 - 2y2 + 20 Where:

x - number of generator engines in thousands.

y - number of drilling engines in thousands.

Required: At what quantities of x and y is revenue maximised in a particular quarter. (5 marks)

c) Define the terms systematic, stratified, multistage, cluster and quota sampling as used in statistical inference.

QUESTION TWO a) Define the following:

i) Markov process. (2 marks) ii) Cyclic chain. (2 marks) iii) Absorbing state. (2 marks) iv) State transition matrix. (2 marks) v) Steady state. (2 marks)

b) In brand switching between different toothpastes named Closed-up CU, Cols-gate CG and Aquas-fresh AF, the state transition matrix below is obtained for a particular month.

From

CU To

CG AF

Required:

CU CG AF

0 5 0 45 0 1

0 2 0 7 0 1

0 1 0 2 0 7

i) If the market share for the different toothpastes were 0.4, 0.3 and 0.3 for CU, CG and AF for January 2002, what will be the market share in February 2002 and March 2002? (6marks)

SHAHIHI


ii) What is the market share equilibrium situation? (4marks)

QUESTION THREE a) Write short notes on the following:

i) Price relatives. (2marks) ii) Fixed base method. (2marks) iii) Chain base method. (2marks) iv) Weighted index number. (2marks) v) Quantity index number. (2marks)

b) The price index for 2001 with base 1985=100 for the listed goods is.

Goods Price Index

X 100 Y 130 Z 180

Find the Arithmetic mean and geometric mean of these three indices. (5marks)

c) In a base period, salaries comprise 50% of selling price of product, materials and overheads 40%, and profits 10%. If salary structure rose by 20% and materials and overhead costs rose by 10% from base period. What must have been the percentage increase in selling price if profits remained 10% of selling

price? (5marks)

QUESTION FOUR a) What is a Venn diagram? How can it be used to determine probability of a given event? (4 marks)

b) Zinco International Corporation had 1,500 employees. In the year 2001, 300 employees got salary increment, 100 were promoted and 50 got both increment and promotion.

Required: i) How many employees got only a promotion? What is that probability? (2 marks) ii) How many employees got neither increment nor promotion? Give that probability (2 marks)

c) i) Explain the meaning of a random variable (2 marks) ii) Zebro Match Manufacturers finds that 5% of the matches in a box are defective. Determine the

probability that out of a box containing 50 match sticks:

a) None will be defective (4 marks) b) If the guarantee is not more than 4 will be defective, what is the probability that the box will

meet guarantee quality? (6 marks)

QUESTION FIVE a) Forecasting is the attempt to predict the future by using qualitative or quantitative means. Discuss the

qualitative and quantitative techniques available for managers in predicting the future. (6marks)

b) i) What are the steps in time-series decomposition? (3marks) ii) Given the following data of production of vehicles by a local car assembly KMA Ltd. Determine

the deseasonalised data and using the forecasting relationship forecast the production in May 2003. Production in „000‟

Year Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec.

2001 7.92 7.81 7.91 7.03 7.25 7.17 5.01 3.9 6.64 7.03 6.88 6.14

2002 4.86 4.48 5.26 5.48 6.42 6.82 4.98 2.45 4.51 6.38 7.55 7.59 (11marks)



SECTION II

QUESTION SIX a) Define a dual problem in relation to the primal problem. (3 marks) b) A fruit/juice kiosk along Lusaka road, VegFresh kiosk has come up with the following formulae of

making natural juice for its industrial area customers.

5 litres lemonade: 2 dozen lemon, 2 kg of sugar, 2 ounces of citric acid and water

5 litres grapefruit: 11/2 kg grape fruit, 11/2 kg of sugar, 1 ounces of citric acid and water

5 litres orangeade: 11/2 dozen of oranges, 11/2 kg of sugar, 1 ounces of citric acid and water The selling prices of the fruit juices per every 5 litre are: Lemonade: Sh. 37.50 Grapefruit: Sh. 40.00 Orangeade: Sh. 42.50

During the cold weather season (May 2002) VegFresh kiosk had in stock 2,500 dozen lemons,2,000 kg grape fruit, 750 dozen oranges, 5,000 kg of sugar and 3,000 ounces of citric acid

Required: i) Formulate the linear programming problem (5 marks) ii) Formulate the dual of this primal (5 marks) iii) The computer solution of the dual was determined to be as follows

(Minimise)

Name Final Value

turnover 83958.3333

Name Final Value

Lemons 0

Grapefruit 7.91666667

Oranges 15.8333333

Sugar 0

Citric acid 18.75

Constraints

Name Cell Value Status Slack

lemonade 37.5 Binding 0

grapefruit 40 Binding 0

orangeade 42.5 Binding 0

Lemons 0 Binding 0

Grapefruit 7.91666667 Not Binding 7.916667

Oranges 15.8333333 Not Binding 15.83333

Sugar 0 Binding 0

Citric acid 18.75 Not Binding 18.75

Sensitivity Report

Final Reduced

Name Value Gradient

Lemons 0 1999.99976

Grapefruit 7.91666667 0

Oranges 15.8333333 0

Sugar 0 1750

Citric acid 18.75 0 SHAHIHI


Constraints

Final Dual Name Value Price

lemonade 37.5 250

grapefruit 40 1333.33333

orangeade 42.5 500

Limits Report

Target Name Value

turnover 83958.3333

Adjustable Lower Target Name Value Limit Result

Y1 0 0 83958.33

Y2 7.91666667 7.916667 83958.33

Y3 15.8333333 15.83333 83958.33

Y4 0 0 83958.33

Y5 18.75 18.75 83958.33

What are the manufacturing quantities in May 2002 that maximises the turnover? And what is the maximum turnover? (4 marks)

iv) Which raw materials are not used up? What are the amounts? (3 marks)

QUESTION SEVEN

a) What is Laplace criterion? How does it differ from Hurwicz criterion? (4marks) b) Sikoka health club specializes in provision of sports/exercise and medical/dietary advice to clients. The service is provided on residential basis and clients reside for whatever number of days that suit their needs. Budget estimates for the year ending 30 June 2003 is as follows:

1) Maximum capacity of center = 50 clients per day for 350 days in the year.

2) Clients will be invoiced at a fee per day. The budgeted occupancy level will vary with the client fee level per day and is estimated at different percentages of maximum capacity as follows:

Client fee per Occupancy level Occupancy % of day in Sh. maximum

capacity

3,600 High 90

4,000 Most likely 75

4,400 Low 60

3) Variable costs are estimated at one of the three levels High 1,900 Most likely 1,700 Low 1,400

4) The range cost levels reflect only the possible effect of the prulus price of goods and services

Required:

i) Summary of budgeted contribution to be earned by Sikoka Health Club for the year ended 30 June 2003 for each of the nine possible outcomes. (10marks)

ii) State the client fee strategy that will result from the use of the maximax, maximin and minimax regret

decision criteria. (6marks)



QUESTION EIGHT MMK Ltd. plans to conduct a survey. The following table shows the tasks involved, the immediately preceding tasks and for each activity duration the most likely estimate (L), the optimistic estimate (O) and the pessimistic estimate (P).

Number of days

Preceding Most likely Optimistic Pessimistic

Activity activity (L) (O) (P)

A - 6 4 8

B - 24 20 40

C A 10 8 24

D B 8 4 12

E D 6 6 6

F B 8 6 10

G C,E,F 20 16 36

H G 6 4 8

I G 4 4 4

J H 10 8 12

K I,J 8 4 24

Using the project evaluation and review technique (PERT) the mean time M and standard deviation , for the duration of each task are estimated from the most likely (L), optimistic (O), pessimistic (P) estimates by using the formulae

M=0.08333(4L+O+P)

=0.08333(P-O)

Required:

a) Compute the mean duration and standard deviation for each task (9marks) b) The project is budgeted to cost Sh.500,000. Actual costs per day are Sh.10,000. By first identifying the critical path from drawing a network diagram can the project be implemented within the budget. (9 marks) c) What is the probability of finishing the project 4 days earlier than the expected duration? (2 marks)

SHAHIHI


PAPER 3

Time Allowed: 3 hours

Answer any THREE questions in SECTION I and TWO questions in SECTION II. Marks allocated to each question are shown at the end of the question. Show all your workings.

SECTION I

QUESTION ONE a) How does differential calculus assist managers in their optimization problems when faced with single

variables? (4marks)

b) The total cost function and demand function of Migoya Construction company (MCC) are as follows:

C = x2 + 16x + 39 Cost in million shillings. P = x2 - 24x + 117 Price per building in million shillings. x - number of buildings

Required: i) Write down the expression of average cost per unit and graph it for x values between x=0 and x=8.

(3marks) ii) Write the expression for total revenue. (2marks) iii) If total revenue is to be maximized, what is the price to be charged? (3marks)

p 1 iv) Elasticity of demand =

x dp dx

Determine the elasticity of demand for the quantity that maximizes total revenue. (5marks) v) What is the price that maximizes profit? (3marks)

QUESTION TWO

a) A Markov process describes the movement among the different states of a system as a function of time.

Give the various steps in using a Markov process for determining a state at (t + 2) and steady state two periods after given state. (6 marks)

b) A major bank BKR Ltd calculates the credit ratings of its credit card customers on a monthly basis. The ratings are poor, good and excellent depending on the payment history. The following matrix shows how the customers change from one category to the other in one month.

To

From

Required:

P oor Good Excellent

P oor 0 .8 0 .18 0 .02

Good 0 .2 0 .75 0 .05

Excellent 0 0 .16 0 .84

i) Interpret the elements 0.84, 0.2, 0.18 and 0.16. (4 marks) ii) Given that in August 2003, from customer base of 100,000 the accounts were classified as

Poor 30,000

Good 50,000

Excellent 20,000

What is expected in October? (10 marks) QUANTITATIVE TECHNIQUES


QUESTION THREE a) Write short notes on the following:

i) Uniform distribution ii) Poisson distribution iii) Normal distribution iv) Beta distribution v) t-distribution (10 marks)

b) A performance test was done before and after training on 12 persons appointed in clerical positions in a cement factory warehouse in Athi River. Marks were awarded on a 10-point scale as follows,

Employee A B C D E F G H I J K L

Before training 4 5 3 7 8 6 5 9 10 6 4 3

After training 5 4 6 8 7 5 9 9 10 6 5 4

Did the training improve performance? (10 marks)

QUESTION FOUR

a) A sample should reduce as much as possible elements of bias since they deprive statistical result of its

representativeness. Define a bias and describe at least four examples of ways that bias is introduced in a sample. (6 marks)

b) A researcher finds that 65% of a random sample of 100 technicians can be classified as highly dissatisfied with their job. With the scale used to measure job satisfaction, the percentage of highly dissatisfied people in all types of occupation has averaged 45%. Can the researcher conclude that a larger proportion of technicians is dissatisfied than is true for all occupations. (Use 5%level of significance) (4 marks)

c) In testing records generated by two sub departments in a large firm, the following were the incorrect records obtained.

Number in sample Mean Standard deviation

Production 10 45 7

Sales 5 55 6

Is the difference in number of incorrect records significant? (10 marks)

QUESTION FIVE a) What is the estimated standard error of a regression equation? (2marks) b) A land resale firm, Shamba Nyingi Limited, sells land around the environs of Nairobi. The size and price

data for a given year are as shown.

Size („000‟ of Price in Sh. square meters) Millions

2.3 4.2

2.4 4.5

3.0 4.8

2.6 4.9

2.6 5.3

3.0 6.2

2.7 7.5

3.6 8.5

2.0 3.5

Required: i) Determine the linear-regression equation of price on size. (8marks) ii) Describe the relationship between size and price. (2marks) iii) What do the equation coefficients represent? (3marks)

SHAHIHI


iv) What is the estimated standard error of the regression equation? (2marks) v) Determine the prize when the size of land is 2,150 square meters. (3marks)

SECTION II

QUESTION SIX a)

i) What is sensitivity analysis on the objective function coefficients? Differentiate this with sensitivity analysis on the right hand side (RHS) constants (4 marks)

ii) Differentiate linear programming from assignment and transportation problems. (4 marks)

b) In a linear programming problem the objective function and constraints were determined to be as follows.

Maximise Z= -x1-x2+3x3-2x4 Profit

Subject to : x 1 3x 2 -x 3 2x 4 7 Resource A

-x 1

-x 1

-2x 2 -4x 4 12 Resource B

4x 2 3x 3 8x 4 10 Resource C

Where xj 0 , j=1,2,3,4

Required: i) By simplex method find the final table. (7 marks) ii) Give the optimal solution and the value of the corresponding objective function (2 marks) iii) For the variable x2, give an interval for their objective function coefficient such that the present

basic solution remains optimal. (3 marks)

QUESTION SEVEN

a) The marketing department of EAA Ltd developed a sales forecasting function for its washing powder and those of competitor Hankol Ltd. EAA Ltd has 3 strategies and Hankol Ltd has 4 strategies. The increase and decrease in quarterly sales revenue for the different combinations of strategies of EAA Ltd and Hankol Ltd are as shown in the payoff matrix:

Hankol Ltd

1 2 3 4

1 5,000 -2,000 12,000 -5,000 EAA Ltd 2 6,000 2,000 7,000 6,000

3 - 2,000 0 - 400 7,000

Required: Using graphical or otherwise determine the strategy available for EAA to pursue. (5marks)

b) A milk processing plant Oldonyo Kesses Ltd in Rift-Valley desires to determine how many kilograms of butter to produce per day to meet demand. Past records indicate the following patterns of demand:

Quantity in Kgs Number of days which demand occurred

15 4

20 16

25 20

30 80

35 40

40 30

45 10 QUANTITATIVE TECHNIQUES


Stock levels are restricted to the range 15 - 45 kilograms (in multiples of 5 kg.) and butter left is disposed by giving the employees working in the factory at the end of the day. Cost of production is Sh.44 and selling price is Sh.50 per Kg.

Required:

i) Construct a payoff table. (10marks) ii) Determine the production alternative that maximises expected profit. (5marks)

QUESTION EIGHT a) In relation to project analysis define the terms:

i) Latest start time. (1mark) ii) Earliest start time. (1mark) iii) Resource leveling. (3marks)

b) A project has 11 activities with the activities preceding constraints, time estimates and work team requirements as shown:

Activity Preceding Days Work team activity

A - 10 5

B A 8 3

C A 5 5

D B 6 4

E D 8 1

F C 7 2

G E,F 4 2

H F 2 1

I F 3 3

J H,I 3 1

K J,G 2 5

- Required: i) Prepare a network and indicate critical path. (6marks) ii) Prepare a time chart and work team requirement. (6marks) iii) Can the work team resource be levelled further? Why? (3marks)

SHAHIHI

Quantitative Techniques Comprehensive Mock Examinations

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